\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

↓

\[\left(0.5 \cdot \cos re\right) \cdot \left({im}^{5} \cdot \left(-\frac{1}{60}\right) - (\left({im}^{3}\right) \cdot \frac{1}{3} + \left(2 \cdot im\right))_*\right)\]

Target

Original	58.2
Target	0.2
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

Initial program 58.2
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
Taylor expanded around 0 0.6
\[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)}\]
Simplified0.6
\[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(-\frac{1}{60}\right) \cdot {im}^{5} - (\left({im}^{3}\right) \cdot \frac{1}{3} + \left(im \cdot 2\right))_*\right)}\]
Final simplification0.6
\[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left({im}^{5} \cdot \left(-\frac{1}{60}\right) - (\left({im}^{3}\right) \cdot \frac{1}{3} + \left(2 \cdot im\right))_*\right)\]

Runtime

herbie shell --seed 2018221 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))

Error

Target

Derivation

Runtime